I cannot think of any references to hanging in the novels, whether humorous or otherwise, but JA did use expressions similar to that in 74 in a couple of earlier letters.

Letter 15 (Brabourne XIV): JA sends good news about their brother Frank's promotion prospects followed by :
"There! I may now finish my letter and go and hang myself, for I am sure I can neither write nor do anything which will not appear insipid to you after this. "

In Letter 32 (Brabourne XXVII) she writes: "I have a great mind not to acknowledge the receipt of your letter, which I have just had the pleasure of reading, because I am so ashamed to compare the sprawling lines of this with it. But if I say all that I have to say, I hope I have no reason to hang myself."

It seems that this was a figure of speech used in a light-hearted way between JA and Cassandra. As I've said in another post it seems to be employed as a bit of dramatic overstatement intended only to amuse. For most of us judicial executions are decidedly not a subject for levity, but this doesn't seem to be the case for JA. I think the context of JA's time may help to explain the difference.

During JA's lifetime the "Bloody Code" still applied, under which more than two hundred offences were punishable by death, some of them trivial by modern standards.

Gibbeting was still carried out whereby the bodies of executed felons were exhibited by the roadside, sometimes for years, in an iron cage. There are quite a few places called "Gibbet Lane" or "Gibbet Hill" in England - one example is not many miles from Chawton, on the London-Portsmouth Road near Petersfield. The practice of gibbeting was abolished in 1834, and hangings were carried out as a public spectacle until 1868.